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The product of the data Pauli Z gates has computational-basis eigenvalueUse the parity circuit $W$ from part (ii), apply the target unitary gate $R=e^{itZ}=\mathrm{diag}(e^{it},e^{-it})$, and then uncompute the parity with $W^{†}$. For every basis input,This equals $e^{it{Z}^{\otimes n}}$ on the data, with the ancilla qubit returned to zero. The compute-phase-uncompute construction therefore gives an exact linear-size circuit:This is a Pauli-string phase by parity computation, an instance of simulation of a computable diagonal Hamiltonian. Part (i) also explains it by $W^{†}(I\otimes Z)W=Z^{\otimes n}\otimes Z$, whose exponential restricts correctly to the target-$|0⟩$ subspace. If no extra line is desired, accumulate parity into the last data qubit, apply $e^{itZ}$ there, and undo the $n-1$ CNOT gates, for $2n-1$ gates. Both constructions implement the global phase as well as the relative phases exactly.

A two-qubit parity phase gate multiplies odd-parity computational vectors by $e^{i\theta }$ and fixes even-parity ones. With $R_{\theta }=\mathrm{diag}(1,e^{i\theta })$,The first controlled-NOT gate computes $x\oplus y$ in the second wire, the phase gate multiplies by $e^{i\theta (x\oplus y)}$, and the last controlled-NOT gate restores the original bits. This compute-phase-uncompute construction leaves the data unchanged except for the desired phases. The more general Pauli-string phase by parity computation uses the same mechanism. In terms of the Pauli Z gate, $G_{\theta }=e^{i\theta /2}\exp [-i\theta (Z\otimes Z)/2]$; the scalar factor ensures equality as a matrix, rather than only up to global phase.